Can $V$ only have well-orderings definable with respect to a parameter?
Of course.
If you start with a model where there is a definable well-ordering, say $V=L$, and you add a single Cohen real $r$ you have that:
- $V=L[r]$, so there is a definable well-ordering with a parameter $r$ (e.g. given two sets in $L[r]$ ask which one has a name appearing first in the order of the ground model, here $L$, that when interpreted with $r$ as the generic give you the two sets).
- Since the Cohen forcing is homogeneous, $L=\mathsf{HOD}^L=\mathsf{HOD}^{L[r]}\neq L[r]$.
So while there is a well-ordering definable from a parameter, which in this case is a real number (read: a subset of $\omega$), there is no such well-ordering which is definable without parameters or from ordinals.
(This argument shows that any set forcing over a model with global well ordering from parameters will also have a global well ordering definable from a parameter. You can violate that with a class forcing, though.)
I just noticed this question, which I find quite interesting.
I thought I'd mention the following related result, which some readers may find interesting:
Theorem. The following are equivalent.
- The universe is HOD of a set: $\exists b\ (V=\text{HOD}(b))$.
- The axiom V=HOD is forceable.
- Somewhere in the generic multiverse, the universe is HOD of a set.
- Somewhere in the generic multiverse, the axiom V=HOD holds.
The proof is contained in my blog post, Being HOD-of-a-set is invariant throughout the generic multiverse.
In particular, it follows that the axiom V=HOD is a switch, in models for which $V=\text{HOD}(b)$, since it can be forced on and then off again as much as you like. If $V=\text{HOD}$ holds, then you can do the forcing in Asaf's answer, adding a Cohen real, and $V\neq\text{HOD}$ in the extension $V[c]$, but then you can force $V=\text{HOD}$ again in a further forcing extension. And furthermore, whenever $V=\text{HOD}(b)$, then you can force $V=\text{HOD}$, and the assertion $\exists b\ V=\text{HOD}(b)$ is invariant by forcing.